of Edinburgh, Session 1881-82. 
409 
radius, and metacarpal of the third finger were got. Their original 
position in the cist with reference to the human remains could not 
be ascertained, as they had been repeatedly changed by visitors. 
The humerus showed the upper epiphysis unanchylosed, so that 
the animal, large as it must have been, was not fully developed. 
4. On a Class of Permanent Symmetric Functions. 
By Thomas Muir, M.A. 
1. By Cauchy the word “ symmetric,” as applied to functions, 
was used in a more general sense than that in which we ordinarily 
use it now, or than that in which it was used before his time. He 
spoke* of “fonctions symetriques permanentes” and “ fonctions 
symetriques alternees we apply the word symmetric only to 
certain of the first of these ; the second we call simply “ alternating 
functions.” Thus the expression 
a 1 b 1 + a 2 b 2 + a 3 b 3 
was called by him a permanent symmetric function, since if oq, a 2J a 3 
are interchanged in order with b 1 , b 2 , b 3 , the function remains un- 
altered. It would be more definite to say that it is a permanent 
symmetric function with respect to a t , a 2 , a 3 and Zq, b 2 , b 3 . Again, 
( a i ~ b^){a 2 — b^)(a 3 — b 3 ) 
he called an alternating symmetric function, since if the same change 
is made the function remains unaltered in magnitude but not in 
sign. As before, for the sake of definiteness, it is better to speak of 
it as an alternating function with respect to a lf a 2 , a 3 and Zq,Z> 2 , b 3 . 
2. Under the latter class of functions it is evident that determi- 
nants may be placed. Thus the determinant 
eq a 2 a 3 
\ \ h 
C 1 C 2 C B 
is an alternating symmetric function with respect to its three rows 
* Journ. de V jfccole polyt. , Cali. xvii. p. 30. 
